A silver prospector is unable to pay his March rent in advance. He owned a bar of

A silver prospector is unable to pay his March rent in advance. He owned a bar of pure silver, 31 inches long, so he made the following arrangement with his landlady. He would cut the bar, he said into smaller pieces. On the first day of March he would give her and inch of the bar, and on each succeeding day he would add another inch to her amount of silver. She would keep this silver as security. At the end of the month, when the prospector expected to be able to pay his rent in full, she would return the pieces to him.March has 31 days, so one way to cut the bar would be to cut it into 31 sections, each an inch long. But since it requires considerable labor to cut the bar the prospector wished to carry out his agreement with the fewest possible number of pieces. For example, he might give the lady an inch piece in the first day, another inch on the second day, then on the third day take back the two inch segments and give her a three inch piece on the third day. Assuming that the portions of the bar are traded back and forth in this fashion, see if you can determine the smallest number of pieces into which the prospector cut his silver bar. Why?